Tyler's and Maronna's M-estimators, as well as their regularized variants, are popular robust methods to estimate the scatter or covariance matrix of a multivariate distribution. In this work, we study the non-asymptotic behavior of these estimators, for data sampled from a distribution that satisfies one of the following properties: 1) independent sub-Gaussian entries, up to a linear transformation; 2) log-concave distributions; 3) distributions satisfying a convex concentration property. Our main contribution is the derivation of tight non-asymptotic concentration bounds of these M-estimators around a suitably scaled version of the data sample covariance matrix. Prior to our work, non-asymptotic bounds were derived only for Elliptical and Gaussian distributions. Our proof uses a variety of tools from non asymptotic random matrix theory and high dimensional geometry. Finally, we illustrate the utility of our results on two examples of practical interest: sparse covariance and sparse precision matrix estimation.

翻译：泰勒和马罗纳M估计量及其正则化变体是估计多元分布散布矩阵或协方差矩阵的常用稳健方法。本研究针对满足以下任一条件分布中采样的数据，探讨这些估计量的非渐近行为：1) 经线性变换后具有独立次高斯分量的数据；2) 对数凹分布；3) 满足凸集中性质的分布。我们的主要贡献在于推导出这些M估计量在数据样本协方差矩阵适当缩放版本周围的紧致非渐近集中界。本研究之前，非渐近界仅适用于椭圆分布和高斯分布。证明过程中使用了非渐近随机矩阵理论和高维几何学的多种工具。最后，通过稀疏协方差与稀疏精度矩阵估计两个实际应用案例展示了结果的实用性。